Vtyjkyuk

I am struggling with the second part of this problem:

Let $A$ be a unital Banach Algebra, and let $x$ and $y$ be elements in A such that that $xy=1_A$ and $yxneq1_A$.

(i) Let $z$ be an element in $A$, such that $|x-z|<frac1$. Show that $z$ is not invertible.

(ii) Let $H$ be an infinite dimensional Hilbert space. Let $Sin B(H)$ be a non-unitary isometry (i.e. $S^*S=I$ and $SS^* neq I$).
Show that $$1=|S|=textrmdist(S,G(B(H))),$$ where $G(B(H))$ is the subset of invertible elements in $B(H)$.

This is the progress, I have made: I have finished the proof of part (i) using the fact that if $|1_A-x|<1$, for some $xin A$, then $x$ is invertible . Further, it is easy to see that $|S|=|S^*|=1$. It therefore follows from part (i) that $textrmdist(S^*, G(B(H))geq 1$. Finally, it is evident that neither $S$ nor $S^*$ are invertible.

Any hints would be appreciated.

We have $|varepsilon I-S|geq1$, since $|I-S|<1$ would imply that $S$ is invertible. So $$1=|S|leq|varepsilon I-S|leq |S|+varepsilon$$ for all $varepsilon>0$. Thus $textdist,(S^*,G(B(H)))leq1+varepsilon$ for all $varepsilon >0$, and thus
$$tag1
textdist,(S,G(B(H)))leq1.
$$
On the other hand, using $(i)$ we have that if $T$ is invertible, then
$$tag2
|S-T|geqfrac1S=1.
$$
Combining $(1)$ and $(2)$, we get $$textdist,(S,G(B(H)))=1.$$